Manin's and Peyre's conjectures on rational points and adelic mixing

Gorodnik, Alex; Maucourant, François; Oh, Hee

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 385-437.

Soit $X$ la compactification merveilleuse d’un groupe semi-simple $𝐆$, connexe, de type adjoint, algébrique défini sur un corps de nombre $K$. Nous démontrons l’asymptotique conjecturée par Manin du nombre de points $K$-rationnels sur $X$ de hauteur plus petite que $T$, lorsque $T\to +\infty$, et construisons de manière explicite une mesure sur $X\left(𝔸\right)$, généralisant celle de Peyre, qui décrit la répartition asymptotique des points rationnels $𝐆\left(K\right)$ sur $X\left(𝔸\right)$. Ce travail repose sur la propriété de mélange de $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$, qui est démontrée avec une estimée de vitesse.

Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$. We prove Manin’s conjecture on the asymptotic (as $T\to \infty$) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X\left(𝔸\right)$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆\left(K\right)$ on $X\left(𝔸\right)$. Our approach is based on the mixing property of $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$ which we obtain with a rate of convergence.

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